Heisenberg Group - The Connection With The Weyl Algebra

The Connection With The Weyl Algebra

The Lie algebra of the Heisenberg group was described above as a Lie algebra of matrices. The Poincaré–Birkhoff–Witt theorem applies to determine the universal enveloping algebra . Among other properties, the universal enveloping algebra is an associative algebra into which injectively imbeds. By Poincaré–Birkhoff–Witt, it is the free vector space generated by the monomials

where the exponents are all non-negative. Thus consists of real polynomials

with the commutation relations

The algebra is closely related to the algebra of differential operators on Rn with polynomial coefficients, since any such operator has a unique representation in the form

This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra W_n is a quotient of . However, this is also easy to see directly from the above representations; viz. by the mapping